For the precision finishing and microfinishing of bores using rotating tools, fine boring heads are known which have a cutting edge which is screwed onto an adaptor, forming a connecting piece between the head and the machine spindle. The cutting edge of the head comes to rest at a random position on the 360.degree. circumference. At the end of the machining of the bore, the machine spindle with the fine boring head is stopped and the tool is withdrawn from the bore in the axial direction. The cutting edge thereby produces a more or less deep groove on the bore wall and in many cases this is undesired. In the case of modern, digitally controlled machine tools, the machine spindles can always be automatically stopped in the same, predetermined radial position. The machine can be programmed in such a way that, prior to withdrawal from the finished bore, the tool can be radially displaced by a short distance, so that the cutting edge is no longer in contact with the bore wall and no groove is formed when the tool is axially withdrawn.